Among all two-dimensional algebras of the second rank with unit e over the field of complex numbers ℂ, we find a semisimple algebra 𝔹0 := {c1e + c2𝜔 : ck 𝜖 ℂ, k = 1, 2}, 𝜔2 = e, containing bases {e1, e2} such that the 𝔹0-valued “analytic” functions Φ(xe1 + ye2), where x and y are real variables, satisfy a homogeneous partial differential equation of the fourth order, which has only simple nonzero characteristics. The set of pairs ({e1, e2},Φ) is described in the explicit form.
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Translated from Ukrains’kyi Matematychnyi Zhurnal,Vol. 73, No. 4, pp. 474–487, April, 2021. Ukrainian DOI: 10.37863/umzh.v73i4.6199.
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Gryshchuk, S.V. Monogenic Functions with Values in Commutative Complex Algebras of the Second Rank with Unit and a Generalized Biharmonic Equation with Simple Nonzero Characteristics. Ukr Math J 73, 556–571 (2021). https://doi.org/10.1007/s11253-021-01943-w
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DOI: https://doi.org/10.1007/s11253-021-01943-w